{"history":[],"article":{"id":"68fb36fe-bd65-4e50-a244-cc59860b885d","title":"Schwarz–Christoffel mapping","slug":"schwarz-christoffel-mapping","content":":::summary\n:::\n\n# \nIn [complex analysis](https://bliptext.com/articles/complex-analysis), a **Schwarz–Christoffel mapping** is a [conformal map](https://bliptext.com/articles/conformal-map) of the [upper half-plane](https://bliptext.com/articles/upper-half-plane) or the complex [unit disk](https://bliptext.com/articles/unit-disk) onto the interior of a [simple polygon](https://bliptext.com/articles/simple-polygon).\nSuch a map is [guaranteed to exist](https://bliptext.com/articles/existence-theorem) by the [Riemann mapping theorem](https://bliptext.com/articles/riemann-mapping-theorem) (stated by [Bernhard Riemann](https://bliptext.com/articles/bernhard-riemann) in 1851); the Schwarz–Christoffel formula provides an explicit construction.\nThey were introduced independently by [Elwin Christoffel](https://bliptext.com/articles/elwin-bruno-christoffel) in 1867 and[ Hermann Schwarz](https://bliptext.com/articles/hermann-amandus-schwarz) in 1869.\nSchwarz–Christoffel mappings are used in [potential theory](https://bliptext.com/articles/potential-theory) and some of its applications, including [minimal surfaces](https://bliptext.com/articles/minimal-surface), [hyperbolic art](https://bliptext.com/articles/poincare-disc-model), and [fluid dynamics](https://bliptext.com/articles/fluid-dynamics).\n\n# Definition\nConsider a polygon in the complex plane.\nThe [Riemann mapping theorem](https://bliptext.com/articles/riemann-mapping-theorem) implies that there is a [biholomorphic](https://bliptext.com/articles/biholomorphic) mapping f from the upper half-plane\nto the interior of the polygon.\nThe function f maps the real axis to the edges of the polygon.\nIf the polygon has interior [angles](https://bliptext.com/articles/angle), then this mapping is given by where K is a [constant](https://bliptext.com/articles/constant-mathematics), and are the values, along the real axis of the \\zeta plane, of points corresponding to the vertices of the polygon in the z plane.\nA transformation of this form is called a Schwarz–Christoffel mapping.\nThe integral can be simplified by mapping the [point at infinity](https://bliptext.com/articles/point-at-infinity) of the \\zeta plane to one of the vertices of the z plane polygon.\nBy doing this, the first factor in the formula becomes constant and so can be absorbed into the constant K. Conventionally, the point at infinity would be mapped to the vertex with angle \\alpha.\nIn practice, to find a mapping to a specific polygon one needs to find the values which generate the correct polygon side lengths.\nThis requires solving a set of nonlinear equations, and in most cases can only be done [numerically](https://bliptext.com/articles/numerical-analysis).\n\n# Example\nConsider a semi-infinite strip in the\nz\n[plane](https://bliptext.com/articles/complex-plane).\nThis may be regarded as a limiting form of a [triangle](https://bliptext.com/articles/triangle) with vertices\nP = 0\n,\nQ = π i\n, and\nR\n(with\nR\nreal), as\nR\ntends to infinity.\nNow\nα = 0\nand\nβ = γ = π/2\nin the limit.\nSuppose we are looking for the mapping\nf\nwith\nf(−1) = Q\n,\nf(1) = P\n, and\nf(∞) = R\n. Then\nf\nis given by\nEvaluation of this integral yields\nwhere\nC\nis a (complex) constant of integration.\nRequiring that\nf(−1) = Q\nand\nf(1) = P\ngives\nC = 0\nand\nK = 1\n. Hence the Schwarz–Christoffel mapping is given by\nThis transformation is sketched below.\n# [Schwarz–Christoffel mapping of the upper half-plane to the semi-infinite strip | upload.wikimedia.org/wikipedia/commons/c/c2/Schwarz-Christoffel///transformation.png]\n\n# Other simple mappings\n\n## Triangle\nA mapping to a plane [triangle](https://bliptext.com/articles/triangle) with interior angles and \\pi(1-a-b) is given by\nwhich can be expressed in terms of [hypergeometric functions](https://bliptext.com/articles/hypergeometric-function), more precisely incomplete [beta functions](https://bliptext.com/articles/beta-function).\n\n## Square\nThe upper half-plane is mapped to the square by\nwhere F is the incomplete [elliptic integral](https://bliptext.com/articles/elliptic-integral) of the first kind.\n\n## General triangle\nThe upper half-plane is mapped to a triangle with circular arcs for edges by the [Schwarz triangle map](https://bliptext.com/articles/schwarz-triangle-map).","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T21:41:39.322Z","updated_at":"2025-01-25T21:41:39.322Z","search_vector":"'christoffel':2 'map':3 'schwarz':1","search_content":" \n \n\n# \nIn [complex analysis](https://bliptext.com/articles/complex-analysis), a **Schwarz–Christoffel mapping** is a [conformal map](https://bliptext.com/articles/conformal-map) of the [upper half-plane](https://bliptext.com/articles/upper-half-plane) or the complex [unit disk](https://bliptext.com/articles/unit-disk) onto the interior of a [simple polygon](https://bliptext.com/articles/simple-polygon).\nSuch a map is [guaranteed to exist](https://bliptext.com/articles/existence-theorem) by the [Riemann mapping theorem](https://bliptext.com/articles/riemann-mapping-theorem) (stated by [Bernhard Riemann](https://bliptext.com/articles/bernhard-riemann) in 1851); the Schwarz–Christoffel formula provides an explicit construction.\nThey were introduced independently by [Elwin Christoffel](https://bliptext.com/articles/elwin-bruno-christoffel) in 1867 and[ Hermann Schwarz](https://bliptext.com/articles/hermann-amandus-schwarz) in 1869.\nSchwarz–Christoffel mappings are used in [potential theory](https://bliptext.com/articles/potential-theory) and some of its applications, including [minimal surfaces](https://bliptext.com/articles/minimal-surface), [hyperbolic art](https://bliptext.com/articles/poincare-disc-model), and [fluid dynamics](https://bliptext.com/articles/fluid-dynamics).\n\n# Definition\nConsider a polygon in the complex plane.\nThe [Riemann mapping theorem](https://bliptext.com/articles/riemann-mapping-theorem) implies that there is a [biholomorphic](https://bliptext.com/articles/biholomorphic) mapping f from the upper half-plane\nto the interior of the polygon.\nThe function f maps the real axis to the edges of the polygon.\nIf the polygon has interior [angles](https://bliptext.com/articles/angle), then this mapping is given by where K is a [constant](https://bliptext.com/articles/constant-mathematics), and are the values, along the real axis of the \\zeta plane, of points corresponding to the vertices of the polygon in the z plane.\nA transformation of this form is called a Schwarz–Christoffel mapping.\nThe integral can be simplified by mapping the [point at infinity](https://bliptext.com/articles/point-at-infinity) of the \\zeta plane to one of the vertices of the z plane polygon.\nBy doing this, the first factor in the formula becomes constant and so can be absorbed into the constant K. Conventionally, the point at infinity would be mapped to the vertex with angle \\alpha.\nIn practice, to find a mapping to a specific polygon one needs to find the values which generate the correct polygon side lengths.\nThis requires solving a set of nonlinear equations, and in most cases can only be done [numerically](https://bliptext.com/articles/numerical-analysis).\n\n# Example\nConsider a semi-infinite strip in the\n z \n[plane](https://bliptext.com/articles/complex-plane).\nThis may be regarded as a limiting form of a [triangle](https://bliptext.com/articles/triangle) with vertices\n P = 0\n,\n Q = π i\n, and\n R \n(with\n R \nreal), as\n R \ntends to infinity.\nNow\nα = 0\nand\nβ = γ = π/2\nin the limit.\nSuppose we are looking for the mapping\n f \nwith\n f (−1) = Q \n,\n f (1) = P \n, and\n f (∞) = R \n. Then\n f \nis given by\nEvaluation of this integral yields\nwhere\n C \nis a (complex) constant of integration.\nRequiring that\n f (−1) = Q \nand\n f (1) = P \ngives\n C = 0\nand\n K = 1\n. Hence the Schwarz–Christoffel mapping is given by\nThis transformation is sketched below.\n# [Schwarz–Christoffel mapping of the upper half-plane to the semi-infinite strip | upload.wikimedia.org/wikipedia/commons/c/c2/Schwarz-Christoffel///transformation.png]\n\n# Other simple mappings\n\n## Triangle\nA mapping to a plane [triangle](https://bliptext.com/articles/triangle) with interior angles and \\pi(1-a-b) is given by\nwhich can be expressed in terms of [hypergeometric functions](https://bliptext.com/articles/hypergeometric-function), more precisely incomplete [beta functions](https://bliptext.com/articles/beta-function).\n\n## Square\nThe upper half-plane is mapped to the square by\nwhere F is the incomplete [elliptic integral](https://bliptext.com/articles/elliptic-integral) of the first kind.\n\n## General triangle\nThe upper half-plane is mapped to a triangle with circular arcs for edges by the [Schwarz triangle map](https://bliptext.com/articles/schwarz-triangle-map).","revisionCount":0},"hasMore":false,"nextCursor":null}

History